Two Price-like equations and two models.

In Model A, the number of offspring follows a binomial distribution with an expected number of offspring of α + β1pi. This means that the model can be summarized as wi = α + β1pi + εi (see Detailed Calculations A.6 at the end of the Appendices for details). For the transition depicted in panels a and c, we generated an offspring generation using Model A, with α = 1 and β1 = 1. Combining the Generalized Price equation in regression form with Model A includes choosing and so that they minimize the sum of squared differences between wi and . In Model B, the number of offspring follows a binomial distribution with an expected number of offspring of α + β1pi + β1pi2, which means that wi = α + β1pi + β1pi2 + εi. For the transition depicted in panels b and d, we generated an offspring generation using Model B, with α = 1, β1 = 0, and β2 = 1. Combining the Generalized Price equation in regression form with Model B includes choosing , and so that they minimize the sum of squared differences between wi and . Reproduction is asexual, so parents and their offspring are always identical, and E(wΔp) = 0 by definition. For both of these models, we started with a population consisting of 2500 parents for each p-score, ranging from 0 to 1 in increments of 0.1. The four panels represent the four combinations of the two Price-like equations and the two datasets. The red lines in all panels represent the estimated fitnesses as a function of the p-score, as implied by the respective Price-like equations. The aim of this example is not to show that the estimated fitnesses from Price-like equation A match the data generated by Model A, and the estimated fitnesses from Price-like equation B match the data generated by Model B, but the estimated fitnesses from Price equation A do not match the data generated by Model B; that part is obvious. The purpose of this example, instead, is to illustrate that both Price-like equations remain identities, also when they are overspecified (panel c) or underspecified (panel b) with respect to the transition between parent and offspring population they are applied to. With underspecification, it is also visually clear that the estimated fitnesses do not match the data, even though Price-like equation A remains an identity, also when combined with data generated by Model B.

Three rules and three models.

This table gives all combinations of the three rules and the three models discussed in the example. All rules indicate the direction of selection correctly for all models. Yellow indicates a combination of a rule and a model, where the rule is more general than is needed for the model. This leads to one or more b’s being 0. These are relatively harmless overspecifications. Red indicates a combination of a rule and a model, where the rule is not general enough (underspecified) for the model. This leads to b’s and c’s that depend on the population state. Terms that depend on the population state are abbreviated as follows: and (see Detailed Calculations B.3 at the end of the Appendices for calculations). Rule 1 is the standard rule for non-social evolution for linear non-social traits. Rule 2 is the classical Hamilton’s rule. Rule 3 is Queller’s rule16, which is a rule that allows for an interaction effect. Rule 1 is nested in Rule 2, which is nested in Rule 3, which can be nested in a more general rules as well.